1. David Kaplan & Heidi Sweetman University of Delaware Two Methodological Perspectives on the Development of Mathematical Competencies in Young Children: An Application of Continuous & Categorical Latent Variable Modeling

2. Topics To Be Covered… Growth mixture modeling (including conventional growth curve modeling) Latent transition analysis A Substantive Example: Math Achievement & ECLS-K

3. Math Achievement in the U.S. Third International Mathematics & Science Study (TIMMS) has led to increased interest in understanding how students develop mathematical competencies Advances in statistical methodologies such as structural equation modeling (SEM) and multilevel modeling now allow for more sophisticated analysis of math competency growth trajectories. Work by Jordan, Hanich & Kaplan (2002) has begun to investigate the shape of early math achievement growth trajectories using these more advanced methodologies

4. Early Childhood Longitudinal Study-Kindergarten (ECLS-K) Longitudinal study of children who began kindergarten in the fall of 1998 Study employed three stage probability sampling to obtain nationally representative sample Sample was freshened in first grade so it is nationally representative of the population of students who began first grade in fall 1999

5. Data Gathering for ECLS-K Data gathered on the entire sample: –Fall kindergarten (fall 1998) –Spring kindergarten (spring 1999) –Spring first grade (spring 2000) –Spring third grade (spring 2002) Additionally, 27% of cohort sub-sampled in fall of first grade (fall 1999) Initial sample included 22,666 students. –Due to attrition, there are 13,698 with data across the four main time points

6. Two Perspectives on Conventional Growth Curve Modeling The Multilevel Modeling Perspective Level 1 represents intra-individual differences in growth over time –Time-varying predictors can be included at level 1 –Level 1 parameters include individual intercepts and slopes that are modeled at level 2 Level 2 represents variation in the intercept and slopes modeled as functions of time-invariant individual characteristics Level 3 represents the parameters of level 2 modeled as a function of a level 3 unit of analysis such as the school or classroom

7. Two Perspectives on Conventional Growth Curve Modeling The Structural Equation Modeling Perspective Measurement portion links repeated measures of an outcome to latent growth factors via a factor analytic specification. Structural Portion links latent growth factors to each other and to individual level predictors Advantages –Flexibility in treating measurement error in the outcomes and predictors –Ability to be extended to latent class models

8. Measurement Portion of Growth Model = a p- dimensional vector measurement intercepts  = a q- dimensional vector of factors Λ = a p x q matrix of factor loadings y i = p-dimensional vector representing the empirical growth record for child i K = p x k matrix of regression coefficients relating the repeated outcomes to a k – dimensional vector of time-varying predictor variables x i  = p-dimensional vector of measurement errors with a p x p covariance matrix Θ p = # of repeated measurements on the ECLS-K math proficiency test q = # of growth factors k = # of time-varying predictors S = # of time-invariant predictors

9. Structural Portion of Growth Model  = a q- dimensional vector that contains the population initial status & growth parameters B = a q x q matrix containing coefficients that relate the latent variables to each other  = a q-dimensional vector of factors Γ = q x s matrix of regression coefficients relating the latent growth factors to an s- dimensional vector of time-invariant predictor variables z  = q-dimensional vector of residuals with covariance matrix Ψ p = # of repeated measurements on the ECLS-K math proficiency test q = # of growth factors k = # of time-varying predictors S = # of time-invariant predictors  = random growth factor allowing growth factors to be related to each and to time- invariant predictors

10. Limitation of Conventional Growth Curve Modeling Conventional growth curve modeling assumes that the manifest growth trajectories are a sample from a single finite population of individuals characterized by a single average status parameter a single average growth rate.

11. Growth Mixture Modeling (GMM) Allows for individual heterogeneity or individual differences in rates of growth Joins conventional growth curve modeling with latent class analysis –under the assumption that there exists a mixture of populations defined by unique trajectory classes Identification of trajectory class membership occurs through latent class analysis –Uncover clusters of individuals who are alike with respect to a set of characteristics measured by a set of categorical outcomes

12. Growth Mixture Model The conventional growth curve model can be rewritten with the subscript c to reflect the presence of trajectory classes

13. The Power of GMM (Assuming the time scores are constant across the cases)  c captures different growth trajectory shapes Relationships between growth parameters in B c are allowed to be class-specific Model allows for differences in measurement error variances ( Θ ) and structural disturbance variances ( Ψ ) across classes Difference classes can show different relationship to a set of covariates z

19. GMM Conclusions Three growth mixture classes were obtained. Adding the poverty indicator yields interesting distinctions among the trajectory classes and could require that the classes be renamed.

20. GMM Conclusions (cont’d) We find a distinct class of high performing children who are above poverty. They come in performing well. Most come in performing similarly, but distinctions emerge over time.

21. GMM Conclusions (cont’d) We might wish to investigate further the middle group of kids – those who are below poverty but performing more like their above poverty counterparts. Who are these kids? Such distinctions are lost in conventional growth curve modeling.

22. Latent Transition Analysis (LTA) LTA examines growth from the perspective of change in qualitative status over time Latent classes are categorical factors arising from the pattern of response frequencies to categorical items Unlike continuous latent variables (factors), categorical latent variables (latent classes) divide individuals into mutually exclusive groups

23. Development of LTA Originally, Latent Class Analysis relied on one single manifest indicator of the latent variable Advances in Latent Class Analysis allowed for multiple manifest categorical indictors of the categorical latent variable –This allowed for the development of LTA –In LTA the arrangement of latent class memberships defines an individual's latent status –This makes the calculation of the probability of moving between or across latent classes over time possible

24. = the probability of response i to item 2 at time t given membership in latent status p = the probability of response i to item 3 at time t given membership in latent status p LTA Model Proportion of individuals Y generating a particular response y δ = proportion of individuals in latent status p at time t = the probability of response i to item 1 at time t given membership in latent status p = the probability of membership in latent status q at time t + 1 given membership in latent status p at time t t = 1 st time of measurement t + 1 = 2 nd time of measurement i’, i’’ = response categories 1, 2…I for 1 st indicator j’, j’’ = response categories 1, 2…J for 2 nd indicator k’, k’’ = response categories 1, 2…K for 3 rd indicator i’, j’, k’ = responses obtained at time 1 i’’, j’’, k;’ = responses obtained at time t + 1 p = latent status at time t q = latent status at time t + 1

25. Latent Class Model = the proportion of individuals in latent class c. = the probability of response i to item 1 at time t given membership in latent status p = the probability of response i to item 2 at time t given membership in latent status p = the probability of response i to item 3 at time t given membership in latent status p Proportion of individuals Y generating a particular response y

26. LTA Example Steps in LTA 1. Separate LCAs for each wave 2. LTA for all waves – calculation of transition probabilities. 3. Addition of poverty variable

27. LTA Example (cont’d) For this analysis, we use data from (1) end of kindergarten, (2) beginning of first, and (3) end of first. We use proficiency levels 3-5. Some estimation problems due to missing data in some cells. Results should be treated with caution.

28. Math Proficiency Levels in ECLS-K Proficiency Level Kindergarten/First Grade Assessment Third Grade Assessment 1 Number & Shape  Identifying some one-digit numerals  Recognizing geometric shapes  Reading all 1 & 2 digit numerals  Demonstrating understanding of place value in integers to hundreds place 2 Relative Size  Recognizing geometric shapes  Using nonstandard units of length to compare the size of objects  Using knowledge of measurement and rate to solve word problems 3 Ordinality & Sequence  One-to-one counting up to 10 objects  Recognizing a sequence of patterns  Recognizing the next number in a sequence  Identifying ordinal position of an object  Recognizing more complex number patterns 4 Add/Subtract  Solving simple addition and subtraction problems 5 Multiply/Divide  Solving simple multiplication and division problems

32. LTA Conclusions 1.Two stable classes found across three waves. 2.Transition probabilities reflect some movement between classes over time. 3.Poverty status strongly relates to class membership but the strength of that relationship appears to change over time.

33. General Conclusions We presented two perspectives on the nature of change over time in math achievement –Growth mixture modeling –Latent transition analysis While both results present a consistent picture of the role of poverty on math achievement, the perspectives are different.

34. General conclusion (cont’d) GMM is concerned with continuous growth and the role of covariates in differentiating growth trajectories. LTA focuses on stage-sequential development over time and focuses on transition probabilities.

35. General conclusions (cont’d) Assuming we can conceive of growth in mathematics (or other academic competencies) as continuous or stage- sequential, value is added by employing both sets of methodologies.